On Ribet’s lemma for GL $$_2$$ modulo prime powers

Abstract

Let \(\rho :G\rightarrow {{\,\textrm{GL}\,}}_2(K)\) be a continuous representation of a compact group G over a complete discretely valued field K with ring of integers \(\mathcal {O}\) and uniformiser \(\pi \) . We prove that \({{\,\textrm{tr}\,}}\rho \) is reducible modulo \(\pi ^n\) if and only if \(\rho \) is reducible modulo \(\pi ^n\) . More precisely, there exist characters \(\chi _1,\chi _2 :G\rightarrow (\mathcal {O}/\pi ^n\mathcal {O})^\times \) such that \(\det (t - \rho (g))\equiv (t-\chi _1(g))(t-\chi _2(g))\pmod {\pi ^n}\) for all \(g\in G\) , if and only if there exists a G-stable lattice \(\Lambda \subseteq K^2\) such that \(\Lambda /\pi ^n\Lambda \) contains a G-invariant, free, rank one \(\mathcal {O}/\pi ^n\mathcal {O}\) -submodule. Our result applies in the case that \(\rho \) is not residually multiplicity-free, in which case it answers a question of Bellaïche and Chenevier (J Algebra 410:501–525, 2014, pp. 524). As an application, we prove an optimal version of Ribet’s lemma, which gives a condition for the existence of a G-stable lattice \(\Lambda \) that realises a non-split extension of \(\chi _2\) by \(\chi _1\) .